CFA Konfirmatorische Faktoranalyse (confirmatory factor analysis)
M.Psy.205
Peter Zezula (pzezula@uni-goettingen.de)
Rmd
Die Konfirmatorische FA liefert Anpassungsindices die Aussagen darüber machen, wie gut das Modell (welche Items bilden welche Faktoren) zu den vorliegenden Daten passt. Hier ein paar Hinweise zu diesen Indices.
\(\chi^2\) und Sig.: Hat vor allem bei großen Stichproben hohe Power, d. h. die Unterschiede zwischen der realen Covarianzmatrix und der aufgrund des Modells reproduzierten Covarianzmatrix werden sig., obwohl sie praktisch unbedeutend sein können. Daher andere Fit-Parameter.
Daumenregeln: Badness of fit index RMSEA:
- RMSEA < 0.05 gute Modellpassung
- 0.05 < RMSEA < 0.08 adäquate/mäßige Modellpassung
- RMSEA > 0.08 schlechte Modellpassung
Üblicherweise wird das 90%-Konfidenzintervall für den RMSEA angegeben! Sog. „Test of close fit“:
- Nullhypothese: RMSEA ≤ 0.05 Nullhypothese kann beibehalten werden, wenn die untere Intervallgrenze des 90%-Konfidenzintervalls kleiner als 0.05 ist.
- andere Quellen (video-tutorial) setzen die Untergrenze des Intervalls etwas höher an (0.07 wird als ok bezeichnet)
Goodness of fit indices:
- Tucker-Lewis NNFI,
- Bentler CFI Daumenregel: sollten größer als .9 sein
Beispiel Emotionale Intelligenz als CFA mit
library(lavaan)
Ansatz: Passt das zwei Faktor Modell (EA Emotional Attention) (EC Emotional Clarity) zu einem gegebenen Datensatz?
Die beiden Skalen ‘emotionale Selbstaufmerksamkeit’ (EA Emotional Attention) sowie ‘Klarheit über eigene Gefühle’ (EC Emotional Clarity) sind theoretisch angenommen und über entsprechende Formulierungen sprachlich umgesetzt. Alle ungeraden Items erfassen ‘emotionale Selbstaufmerksamkeit’ (EA), alle geraden Items ‘Klarheit über eigene Gefühle’ (EC).
Die Items sind:
- i1 EA Ich denke über meine Gefühle nach.
- i2 EC Ich kann meine Gefühle benennen.
- i3 EA Ich schenke meinen Gefühlen Aufmerksamkeit.
- i4 EC Ich bin mir im unklaren darüber, was ich fühle.
- i5 EA Ich beschäftige mich mit meinen Gefühlen.
- i6 EC Ich habe Schwierigkeiten, meine Gefühle zu beschreiben.
- i7 EA Ich denke darüber nach, wie ich mich fühle.
- i8 EC Ich weiß, was ich fühle.
- i9 EA Ich beobachte meine Gefühle.
- i10 EC Ich habe Schwierigkeiten, meinen Gefühlen einen Namen zu geben.
- i11 EA Ich ache darauf, wie ich mich fühle.
- i12 EC Ich bin mir unsicher, was ich eigentlich fühle.
Die Items sind skaliert von 1 bis 4 (fast nie/manchmal/oft/fast immer)
# get data
ddf <- read.delim("https://md.psych.bio.uni-goettingen.de/mv/data/virt/v_ei.txt")
head(ddf)## i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12
## 1 4 3 4 2 3 1 4 3 3 1 3 3
## 2 4 3 3 1 4 1 4 4 4 1 3 2
## 3 4 3 4 2 4 2 4 2 4 1 3 3
## 4 3 3 2 2 1 2 4 3 1 2 2 3
## 5 4 4 4 4 4 1 4 4 4 2 3 1
## 6 2 4 3 2 4 2 2 3 3 2 3 2# get cov matrix
t.cov <- cov(ddf)
# and correlation matrix
#cor(ddf)
require(lavaan)## Lade nötiges Paket: lavaan## This is lavaan 0.6-11
## lavaan is FREE software! Please report any bugs.m.ei <- '
EA =~ i1 + i3 + i5 + i7 + i9 + i11
EC =~ i2 + i4 + i6 + i8 + i10 + i12
'
# fit the an orthogonal model
fit.o <- cfa(m.ei, data=ddf, orthogonal=T)
# fit a model that allows correlation of factors
fit.c <- cfa(m.ei, data=ddf, orthogonal=F)
# display summary output
summary(fit.o, fit.measures=TRUE)## lavaan 0.6-11 ended normally after 27 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 24
##
## Number of observations 100
##
## Model Test User Model:
##
## Test statistic 69.883
## Degrees of freedom 54
## P-value (Chi-square) 0.072
##
## Model Test Baseline Model:
##
## Test statistic 397.852
## Degrees of freedom 66
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.952
## Tucker-Lewis Index (TLI) 0.942
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1440.660
## Loglikelihood unrestricted model (H1) -1405.718
##
## Akaike (AIC) 2929.320
## Bayesian (BIC) 2991.844
## Sample-size adjusted Bayesian (BIC) 2916.046
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.054
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.088
## P-value RMSEA <= 0.05 0.403
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.111
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## EA =~
## i1 1.000
## i3 0.918 0.234 3.932 0.000
## i5 0.773 0.230 3.358 0.001
## i7 0.465 0.209 2.230 0.026
## i9 0.976 0.246 3.961 0.000
## i11 0.705 0.192 3.675 0.000
## EC =~
## i2 1.000
## i4 -0.914 0.131 -6.981 0.000
## i6 -0.940 0.131 -7.154 0.000
## i8 0.987 0.142 6.970 0.000
## i10 -1.030 0.135 -7.649 0.000
## i12 -0.868 0.126 -6.904 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## EA ~~
## EC 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .i1 0.550 0.102 5.377 0.000
## .i3 0.481 0.088 5.438 0.000
## .i5 0.677 0.108 6.260 0.000
## .i7 0.776 0.114 6.817 0.000
## .i9 0.518 0.097 5.359 0.000
## .i11 0.398 0.067 5.917 0.000
## .i2 0.384 0.069 5.597 0.000
## .i4 0.454 0.075 6.034 0.000
## .i6 0.437 0.074 5.931 0.000
## .i8 0.532 0.088 6.040 0.000
## .i10 0.394 0.071 5.547 0.000
## .i12 0.426 0.070 6.076 0.000
## EA 0.306 0.114 2.684 0.007
## EC 0.550 0.127 4.313 0.000summary(fit.c, fit.measures=TRUE)## lavaan 0.6-11 ended normally after 30 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 25
##
## Number of observations 100
##
## Model Test User Model:
##
## Test statistic 61.027
## Degrees of freedom 53
## P-value (Chi-square) 0.210
##
## Model Test Baseline Model:
##
## Test statistic 397.852
## Degrees of freedom 66
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.976
## Tucker-Lewis Index (TLI) 0.970
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1436.232
## Loglikelihood unrestricted model (H1) -1405.718
##
## Akaike (AIC) 2922.463
## Bayesian (BIC) 2987.593
## Sample-size adjusted Bayesian (BIC) 2908.636
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.039
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.077
## P-value RMSEA <= 0.05 0.642
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.055
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## EA =~
## i1 1.000
## i3 0.927 0.234 3.963 0.000
## i5 0.799 0.233 3.428 0.001
## i7 0.492 0.211 2.328 0.020
## i9 0.996 0.248 4.017 0.000
## i11 0.715 0.193 3.704 0.000
## EC =~
## i2 1.000
## i4 -0.904 0.130 -6.949 0.000
## i6 -0.932 0.130 -7.151 0.000
## i8 0.995 0.140 7.090 0.000
## i10 -1.029 0.133 -7.713 0.000
## i12 -0.861 0.125 -6.897 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## EA ~~
## EC 0.151 0.060 2.513 0.012
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .i1 0.560 0.101 5.549 0.000
## .i3 0.484 0.087 5.559 0.000
## .i5 0.670 0.107 6.268 0.000
## .i7 0.770 0.113 6.808 0.000
## .i9 0.516 0.095 5.431 0.000
## .i11 0.398 0.067 5.984 0.000
## .i2 0.381 0.068 5.602 0.000
## .i4 0.462 0.076 6.086 0.000
## .i6 0.441 0.074 5.973 0.000
## .i8 0.520 0.087 6.009 0.000
## .i10 0.391 0.070 5.555 0.000
## .i12 0.430 0.070 6.113 0.000
## EA 0.297 0.111 2.674 0.007
## EC 0.553 0.127 4.336 0.000# get modification indices also
summary(fit.o, fit.measures = TRUE, modindices = TRUE)## lavaan 0.6-11 ended normally after 27 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 24
##
## Number of observations 100
##
## Model Test User Model:
##
## Test statistic 69.883
## Degrees of freedom 54
## P-value (Chi-square) 0.072
##
## Model Test Baseline Model:
##
## Test statistic 397.852
## Degrees of freedom 66
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.952
## Tucker-Lewis Index (TLI) 0.942
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1440.660
## Loglikelihood unrestricted model (H1) -1405.718
##
## Akaike (AIC) 2929.320
## Bayesian (BIC) 2991.844
## Sample-size adjusted Bayesian (BIC) 2916.046
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.054
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.088
## P-value RMSEA <= 0.05 0.403
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.111
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## EA =~
## i1 1.000
## i3 0.918 0.234 3.932 0.000
## i5 0.773 0.230 3.358 0.001
## i7 0.465 0.209 2.230 0.026
## i9 0.976 0.246 3.961 0.000
## i11 0.705 0.192 3.675 0.000
## EC =~
## i2 1.000
## i4 -0.914 0.131 -6.981 0.000
## i6 -0.940 0.131 -7.154 0.000
## i8 0.987 0.142 6.970 0.000
## i10 -1.030 0.135 -7.649 0.000
## i12 -0.868 0.126 -6.904 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## EA ~~
## EC 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .i1 0.550 0.102 5.377 0.000
## .i3 0.481 0.088 5.438 0.000
## .i5 0.677 0.108 6.260 0.000
## .i7 0.776 0.114 6.817 0.000
## .i9 0.518 0.097 5.359 0.000
## .i11 0.398 0.067 5.917 0.000
## .i2 0.384 0.069 5.597 0.000
## .i4 0.454 0.075 6.034 0.000
## .i6 0.437 0.074 5.931 0.000
## .i8 0.532 0.088 6.040 0.000
## .i10 0.394 0.071 5.547 0.000
## .i12 0.426 0.070 6.076 0.000
## EA 0.306 0.114 2.684 0.007
## EC 0.550 0.127 4.313 0.000
##
## Modification Indices:
##
## lhs op rhs mi epc sepc.lv sepc.all sepc.nox
## 27 EA ~~ EC 8.407 0.151 0.369 0.369 0.369
## 28 EA =~ i2 0.693 0.122 0.068 0.070 0.070
## 29 EA =~ i4 0.336 0.090 0.050 0.052 0.052
## 30 EA =~ i6 0.002 0.007 0.004 0.004 0.004
## 31 EA =~ i8 4.259 0.346 0.192 0.186 0.186
## 32 EA =~ i10 0.627 -0.118 -0.065 -0.066 -0.066
## 33 EA =~ i12 0.195 0.066 0.037 0.040 0.040
## 34 EC =~ i1 0.136 0.043 0.032 0.035 0.035
## 35 EC =~ i3 0.367 0.066 0.049 0.057 0.057
## 36 EC =~ i5 0.824 0.112 0.083 0.090 0.090
## 37 EC =~ i7 0.966 0.126 0.094 0.102 0.102
## 38 EC =~ i9 1.042 0.116 0.086 0.096 0.096
## 39 EC =~ i11 0.341 0.056 0.042 0.056 0.056
## 40 i1 ~~ i3 0.006 -0.006 -0.006 -0.012 -0.012
## 41 i1 ~~ i5 3.456 -0.148 -0.148 -0.243 -0.243
## 42 i1 ~~ i7 0.055 0.018 0.018 0.028 0.028
## 43 i1 ~~ i9 3.151 0.149 0.149 0.278 0.278
## 44 i1 ~~ i11 0.087 -0.019 -0.019 -0.041 -0.041
## 45 i1 ~~ i2 0.733 0.047 0.047 0.103 0.103
## 46 i1 ~~ i4 0.569 0.044 0.044 0.088 0.088
## 47 i1 ~~ i6 5.979 -0.141 -0.141 -0.288 -0.288
## 48 i1 ~~ i8 0.670 -0.052 -0.052 -0.096 -0.096
## 49 i1 ~~ i10 2.326 0.086 0.086 0.185 0.185
## 50 i1 ~~ i12 0.036 -0.011 -0.011 -0.022 -0.022
## 51 i3 ~~ i5 0.261 0.038 0.038 0.066 0.066
## 52 i3 ~~ i7 0.316 -0.041 -0.041 -0.066 -0.066
## 53 i3 ~~ i9 0.209 -0.035 -0.035 -0.071 -0.071
## 54 i3 ~~ i11 0.188 0.026 0.026 0.060 0.060
## 55 i3 ~~ i2 0.157 -0.021 -0.021 -0.048 -0.048
## 56 i3 ~~ i4 0.560 0.041 0.041 0.087 0.087
## 57 i3 ~~ i6 0.178 0.023 0.023 0.050 0.050
## 58 i3 ~~ i8 1.210 0.065 0.065 0.128 0.128
## 59 i3 ~~ i10 4.178 -0.107 -0.107 -0.247 -0.247
## 60 i3 ~~ i12 1.584 0.066 0.066 0.146 0.146
## 61 i5 ~~ i7 0.359 0.047 0.047 0.065 0.065
## 62 i5 ~~ i9 0.060 -0.019 -0.019 -0.032 -0.032
## 63 i5 ~~ i11 2.180 0.093 0.093 0.179 0.179
## 64 i5 ~~ i2 0.230 0.028 0.028 0.055 0.055
## 65 i5 ~~ i4 0.055 -0.014 -0.014 -0.026 -0.026
## 66 i5 ~~ i6 2.314 0.093 0.093 0.171 0.171
## 67 i5 ~~ i8 5.745 0.160 0.160 0.267 0.267
## 68 i5 ~~ i10 0.235 0.029 0.029 0.056 0.056
## 69 i5 ~~ i12 0.016 0.008 0.008 0.014 0.014
## 70 i7 ~~ i9 0.001 0.003 0.003 0.004 0.004
## 71 i7 ~~ i11 0.037 -0.012 -0.012 -0.022 -0.022
## 72 i7 ~~ i2 2.317 0.093 0.093 0.170 0.170
## 73 i7 ~~ i4 0.082 -0.018 -0.018 -0.031 -0.031
## 74 i7 ~~ i6 1.179 0.069 0.069 0.118 0.118
## 75 i7 ~~ i8 0.517 0.050 0.050 0.078 0.078
## 76 i7 ~~ i10 0.163 -0.025 -0.025 -0.045 -0.045
## 77 i7 ~~ i12 1.304 0.071 0.071 0.123 0.123
## 78 i9 ~~ i11 1.495 -0.078 -0.078 -0.171 -0.171
## 79 i9 ~~ i2 0.000 0.000 0.000 0.001 0.001
## 80 i9 ~~ i4 0.067 0.015 0.015 0.030 0.030
## 81 i9 ~~ i6 0.586 -0.043 -0.043 -0.090 -0.090
## 82 i9 ~~ i8 0.046 0.013 0.013 0.025 0.025
## 83 i9 ~~ i10 0.037 -0.011 -0.011 -0.023 -0.023
## 84 i9 ~~ i12 0.001 -0.002 -0.002 -0.004 -0.004
## 85 i11 ~~ i2 0.267 -0.024 -0.024 -0.061 -0.061
## 86 i11 ~~ i4 0.765 -0.042 -0.042 -0.099 -0.099
## 87 i11 ~~ i6 2.248 0.071 0.071 0.172 0.172
## 88 i11 ~~ i8 0.000 0.000 0.000 0.000 0.000
## 89 i11 ~~ i10 0.309 -0.026 -0.026 -0.065 -0.065
## 90 i11 ~~ i12 1.516 -0.057 -0.057 -0.139 -0.139
## 91 i2 ~~ i4 3.914 0.108 0.108 0.260 0.260
## 92 i2 ~~ i6 1.208 0.060 0.060 0.147 0.147
## 93 i2 ~~ i8 0.465 0.040 0.040 0.089 0.089
## 94 i2 ~~ i10 0.503 -0.039 -0.039 -0.101 -0.101
## 95 i2 ~~ i12 2.769 -0.088 -0.088 -0.217 -0.217
## 96 i4 ~~ i6 8.883 0.166 0.166 0.373 0.373
## 97 i4 ~~ i8 1.727 0.079 0.079 0.162 0.162
## 98 i4 ~~ i10 0.226 -0.027 -0.027 -0.063 -0.063
## 99 i4 ~~ i12 0.982 0.053 0.053 0.121 0.121
## 100 i6 ~~ i8 1.314 0.069 0.069 0.143 0.143
## 101 i6 ~~ i10 0.550 0.041 0.041 0.100 0.100
## 102 i6 ~~ i12 2.153 -0.079 -0.079 -0.182 -0.182
## 103 i8 ~~ i10 0.674 -0.050 -0.050 -0.108 -0.108
## 104 i8 ~~ i12 0.679 -0.048 -0.048 -0.101 -0.101
## 105 i10 ~~ i12 3.658 -0.103 -0.103 -0.251 -0.251summary(fit.c, fit.measures = TRUE, modindices = TRUE)## lavaan 0.6-11 ended normally after 30 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 25
##
## Number of observations 100
##
## Model Test User Model:
##
## Test statistic 61.027
## Degrees of freedom 53
## P-value (Chi-square) 0.210
##
## Model Test Baseline Model:
##
## Test statistic 397.852
## Degrees of freedom 66
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.976
## Tucker-Lewis Index (TLI) 0.970
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1436.232
## Loglikelihood unrestricted model (H1) -1405.718
##
## Akaike (AIC) 2922.463
## Bayesian (BIC) 2987.593
## Sample-size adjusted Bayesian (BIC) 2908.636
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.039
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.077
## P-value RMSEA <= 0.05 0.642
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.055
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## EA =~
## i1 1.000
## i3 0.927 0.234 3.963 0.000
## i5 0.799 0.233 3.428 0.001
## i7 0.492 0.211 2.328 0.020
## i9 0.996 0.248 4.017 0.000
## i11 0.715 0.193 3.704 0.000
## EC =~
## i2 1.000
## i4 -0.904 0.130 -6.949 0.000
## i6 -0.932 0.130 -7.151 0.000
## i8 0.995 0.140 7.090 0.000
## i10 -1.029 0.133 -7.713 0.000
## i12 -0.861 0.125 -6.897 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## EA ~~
## EC 0.151 0.060 2.513 0.012
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .i1 0.560 0.101 5.549 0.000
## .i3 0.484 0.087 5.559 0.000
## .i5 0.670 0.107 6.268 0.000
## .i7 0.770 0.113 6.808 0.000
## .i9 0.516 0.095 5.431 0.000
## .i11 0.398 0.067 5.984 0.000
## .i2 0.381 0.068 5.602 0.000
## .i4 0.462 0.076 6.086 0.000
## .i6 0.441 0.074 5.973 0.000
## .i8 0.520 0.087 6.009 0.000
## .i10 0.391 0.070 5.555 0.000
## .i12 0.430 0.070 6.113 0.000
## EA 0.297 0.111 2.674 0.007
## EC 0.553 0.127 4.336 0.000
##
## Modification Indices:
##
## lhs op rhs mi epc sepc.lv sepc.all sepc.nox
## 28 EA =~ i2 0.103 0.054 0.030 0.031 0.031
## 29 EA =~ i4 1.074 0.185 0.101 0.105 0.105
## 30 EA =~ i6 0.299 0.096 0.053 0.055 0.055
## 31 EA =~ i8 3.110 0.336 0.183 0.177 0.177
## 32 EA =~ i10 0.071 -0.046 -0.025 -0.025 -0.025
## 33 EA =~ i12 0.829 0.157 0.085 0.093 0.093
## 34 EC =~ i1 0.254 -0.070 -0.052 -0.056 -0.056
## 35 EC =~ i3 0.049 -0.029 -0.021 -0.025 -0.025
## 36 EC =~ i5 0.138 0.053 0.039 0.042 0.042
## 37 EC =~ i7 0.512 0.103 0.077 0.084 0.084
## 38 EC =~ i9 0.034 0.025 0.019 0.021 0.021
## 39 EC =~ i11 0.012 -0.012 -0.009 -0.012 -0.012
## 40 i1 ~~ i3 0.010 0.008 0.008 0.015 0.015
## 41 i1 ~~ i5 3.273 -0.141 -0.141 -0.230 -0.230
## 42 i1 ~~ i7 0.028 0.013 0.013 0.020 0.020
## 43 i1 ~~ i9 3.099 0.140 0.140 0.260 0.260
## 44 i1 ~~ i11 0.030 -0.011 -0.011 -0.023 -0.023
## 45 i1 ~~ i2 0.512 0.040 0.040 0.086 0.086
## 46 i1 ~~ i4 0.788 0.052 0.052 0.103 0.103
## 47 i1 ~~ i6 5.159 -0.132 -0.132 -0.265 -0.265
## 48 i1 ~~ i8 0.782 -0.056 -0.056 -0.103 -0.103
## 49 i1 ~~ i10 2.712 0.093 0.093 0.198 0.198
## 50 i1 ~~ i12 0.002 -0.003 -0.003 -0.006 -0.006
## 51 i3 ~~ i5 0.200 0.032 0.032 0.057 0.057
## 52 i3 ~~ i7 0.413 -0.046 -0.046 -0.075 -0.075
## 53 i3 ~~ i9 0.177 -0.031 -0.031 -0.062 -0.062
## 54 i3 ~~ i11 0.215 0.027 0.027 0.062 0.062
## 55 i3 ~~ i2 0.306 -0.028 -0.028 -0.066 -0.066
## 56 i3 ~~ i4 0.804 0.049 0.049 0.104 0.104
## 57 i3 ~~ i6 0.300 0.030 0.030 0.064 0.064
## 58 i3 ~~ i8 1.020 0.059 0.059 0.118 0.118
## 59 i3 ~~ i10 3.523 -0.098 -0.098 -0.226 -0.226
## 60 i3 ~~ i12 1.933 0.073 0.073 0.161 0.161
## 61 i5 ~~ i7 0.255 0.040 0.040 0.055 0.055
## 62 i5 ~~ i9 0.127 -0.027 -0.027 -0.046 -0.046
## 63 i5 ~~ i11 1.974 0.087 0.087 0.169 0.169
## 64 i5 ~~ i2 0.136 0.021 0.021 0.043 0.043
## 65 i5 ~~ i4 0.018 -0.008 -0.008 -0.015 -0.015
## 66 i5 ~~ i6 2.570 0.098 0.098 0.180 0.180
## 67 i5 ~~ i8 5.538 0.155 0.155 0.263 0.263
## 68 i5 ~~ i10 0.370 0.036 0.036 0.070 0.070
## 69 i5 ~~ i12 0.049 0.013 0.013 0.024 0.024
## 70 i7 ~~ i9 0.007 -0.006 -0.006 -0.010 -0.010
## 71 i7 ~~ i11 0.073 -0.017 -0.017 -0.030 -0.030
## 72 i7 ~~ i2 2.172 0.089 0.089 0.165 0.165
## 73 i7 ~~ i4 0.059 -0.016 -0.016 -0.026 -0.026
## 74 i7 ~~ i6 1.253 0.071 0.071 0.122 0.122
## 75 i7 ~~ i8 0.450 0.046 0.046 0.073 0.073
## 76 i7 ~~ i10 0.118 -0.021 -0.021 -0.039 -0.039
## 77 i7 ~~ i12 1.391 0.073 0.073 0.127 0.127
## 78 i9 ~~ i11 1.462 -0.075 -0.075 -0.165 -0.165
## 79 i9 ~~ i2 0.020 -0.008 -0.008 -0.017 -0.017
## 80 i9 ~~ i4 0.160 0.023 0.023 0.047 0.047
## 81 i9 ~~ i6 0.417 -0.036 -0.036 -0.076 -0.076
## 82 i9 ~~ i8 0.014 0.007 0.007 0.014 0.014
## 83 i9 ~~ i10 0.002 -0.002 -0.002 -0.005 -0.005
## 84 i9 ~~ i12 0.010 0.006 0.006 0.012 0.012
## 85 i11 ~~ i2 0.402 -0.029 -0.029 -0.074 -0.074
## 86 i11 ~~ i4 0.553 -0.036 -0.036 -0.084 -0.084
## 87 i11 ~~ i6 2.494 0.075 0.075 0.180 0.180
## 88 i11 ~~ i8 0.005 -0.004 -0.004 -0.008 -0.008
## 89 i11 ~~ i10 0.191 -0.020 -0.020 -0.051 -0.051
## 90 i11 ~~ i12 1.235 -0.052 -0.052 -0.125 -0.125
## 91 i2 ~~ i4 3.462 0.101 0.101 0.242 0.242
## 92 i2 ~~ i6 1.109 0.057 0.057 0.139 0.139
## 93 i2 ~~ i8 0.218 0.027 0.027 0.061 0.061
## 94 i2 ~~ i10 0.363 -0.033 -0.033 -0.086 -0.086
## 95 i2 ~~ i12 2.844 -0.088 -0.088 -0.218 -0.218
## 96 i4 ~~ i6 9.440 0.171 0.171 0.379 0.379
## 97 i4 ~~ i8 1.816 0.081 0.081 0.165 0.165
## 98 i4 ~~ i10 0.136 -0.021 -0.021 -0.048 -0.048
## 99 i4 ~~ i12 1.272 0.061 0.061 0.136 0.136
## 100 i6 ~~ i8 1.527 0.074 0.074 0.154 0.154
## 101 i6 ~~ i10 0.604 0.043 0.043 0.104 0.104
## 102 i6 ~~ i12 1.710 -0.070 -0.070 -0.161 -0.161
## 103 i8 ~~ i10 0.372 -0.036 -0.036 -0.081 -0.081
## 104 i8 ~~ i12 0.568 -0.044 -0.044 -0.092 -0.092
## 105 i10 ~~ i12 3.428 -0.099 -0.099 -0.241 -0.241# get standardized indices
standardizedSolution(fit.o) ## lhs op rhs est.std se z pvalue ci.lower ci.upper
## 1 EA =~ i1 0.598 0.091 6.553 0.000 0.419 0.777
## 2 EA =~ i3 0.591 0.092 6.444 0.000 0.411 0.771
## 3 EA =~ i5 0.461 0.101 4.565 0.000 0.263 0.659
## 4 EA =~ i7 0.281 0.112 2.503 0.012 0.061 0.500
## 5 EA =~ i9 0.600 0.091 6.584 0.000 0.421 0.779
## 6 EA =~ i11 0.526 0.096 5.458 0.000 0.337 0.715
## 7 EC =~ i2 0.767 0.050 15.202 0.000 0.668 0.866
## 8 EC =~ i4 -0.709 0.058 -12.191 0.000 -0.823 -0.595
## 9 EC =~ i6 -0.726 0.056 -12.946 0.000 -0.835 -0.616
## 10 EC =~ i8 0.708 0.058 12.142 0.000 0.594 0.823
## 11 EC =~ i10 -0.772 0.050 -15.519 0.000 -0.870 -0.675
## 12 EC =~ i12 -0.702 0.059 -11.873 0.000 -0.818 -0.586
## 13 i1 ~~ i1 0.642 0.109 5.882 0.000 0.428 0.856
## 14 i3 ~~ i3 0.651 0.108 5.998 0.000 0.438 0.863
## 15 i5 ~~ i5 0.787 0.093 8.442 0.000 0.604 0.970
## 16 i7 ~~ i7 0.921 0.063 14.639 0.000 0.798 1.045
## 17 i9 ~~ i9 0.640 0.109 5.849 0.000 0.425 0.854
## 18 i11 ~~ i11 0.723 0.101 7.138 0.000 0.525 0.922
## 19 i2 ~~ i2 0.411 0.077 5.310 0.000 0.259 0.563
## 20 i4 ~~ i4 0.497 0.083 6.020 0.000 0.335 0.659
## 21 i6 ~~ i6 0.474 0.081 5.823 0.000 0.314 0.633
## 22 i8 ~~ i8 0.498 0.083 6.033 0.000 0.337 0.660
## 23 i10 ~~ i10 0.403 0.077 5.243 0.000 0.253 0.554
## 24 i12 ~~ i12 0.507 0.083 6.108 0.000 0.344 0.670
## 25 EA ~~ EA 1.000 0.000 NA NA 1.000 1.000
## 26 EC ~~ EC 1.000 0.000 NA NA 1.000 1.000
## 27 EA ~~ EC 0.000 0.000 NA NA 0.000 0.000standardizedSolution(fit.c) ## lhs op rhs est.std se z pvalue ci.lower ci.upper
## 1 EA =~ i1 0.589 0.090 6.533 0.000 0.412 0.765
## 2 EA =~ i3 0.587 0.090 6.513 0.000 0.411 0.764
## 3 EA =~ i5 0.469 0.099 4.733 0.000 0.275 0.664
## 4 EA =~ i7 0.292 0.111 2.643 0.008 0.075 0.509
## 5 EA =~ i9 0.603 0.089 6.770 0.000 0.428 0.777
## 6 EA =~ i11 0.525 0.095 5.530 0.000 0.339 0.711
## 7 EC =~ i2 0.769 0.050 15.398 0.000 0.672 0.867
## 8 EC =~ i4 -0.703 0.059 -11.958 0.000 -0.818 -0.588
## 9 EC =~ i6 -0.722 0.056 -12.813 0.000 -0.832 -0.612
## 10 EC =~ i8 0.716 0.057 12.545 0.000 0.604 0.828
## 11 EC =~ i10 -0.774 0.049 -15.702 0.000 -0.871 -0.678
## 12 EC =~ i12 -0.698 0.059 -11.750 0.000 -0.815 -0.582
## 13 i1 ~~ i1 0.654 0.106 6.163 0.000 0.446 0.861
## 14 i3 ~~ i3 0.655 0.106 6.182 0.000 0.447 0.863
## 15 i5 ~~ i5 0.780 0.093 8.370 0.000 0.597 0.962
## 16 i7 ~~ i7 0.915 0.065 14.159 0.000 0.788 1.041
## 17 i9 ~~ i9 0.637 0.107 5.930 0.000 0.426 0.847
## 18 i11 ~~ i11 0.724 0.100 7.263 0.000 0.529 0.920
## 19 i2 ~~ i2 0.408 0.077 5.304 0.000 0.257 0.559
## 20 i4 ~~ i4 0.505 0.083 6.111 0.000 0.343 0.668
## 21 i6 ~~ i6 0.479 0.081 5.885 0.000 0.319 0.638
## 22 i8 ~~ i8 0.487 0.082 5.953 0.000 0.327 0.647
## 23 i10 ~~ i10 0.400 0.076 5.242 0.000 0.251 0.550
## 24 i12 ~~ i12 0.512 0.083 6.169 0.000 0.349 0.675
## 25 EA ~~ EA 1.000 0.000 NA NA 1.000 1.000
## 26 EC ~~ EC 1.000 0.000 NA NA 1.000 1.000
## 27 EA ~~ EC 0.372 0.114 3.261 0.001 0.149 0.596# model test
anova(fit.o, fit.c)## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## fit.c 53 2922.5 2987.6 61.027
## fit.o 54 2929.3 2991.8 69.883 8.8564 1 0.002921 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# qgraph is missing - to fix
# we plot the solution using library(semPlot)
# require(semPlot)
# path plot with unstandardized estimates
# semPlot::semPaths(fit.o, 'est')
# semPlot::semPaths(fit.c, 'est')
# path plot with standardized estimates
# semPaths(fit.o, 'est', 'std')
# semPaths(fit.c, 'est', 'std')
semPlot::semPaths(fit.c, "std")
Aufruf des gefitteten Modells fit gibt nur den globalen
Signifikanztest aus, ob die reproduzierte und die originale
Covarianzmatrizen signifikant unterschiedlich sind. Bei sehr hohem n
werden auch sehr kleine Unterschiede signifikant. Im Beispiel wird die
Covarianzmatrix wird sehr gut reproduziert - die Unterschiede könnten
auch Zufallsunterschiede sein.
Summary des Modells summary(fit) gibt noch die
Koeffizienten mit aus (unstandardisiert). Die standardisierten
Koeffizienten, die auch in der Grafik verwendet werden, erhält man über
den Befehl standardizedSolution(fit). Für alle Indikatoren
wird das Ladungsgewicht ausgegeben, das aussagt, wieviel Varianz der
Variablen durch den Faktor erklärt wird (entspricht der Kommunalität).
Dieser Anteil wird auf Signifikanz geprüft.
Das Flag fit.measures in
summary(fit, fit.measures=TRUE) fordert ausführlichere
Details zur Modellanpassung an.
Neben dem exakten Übereinstimmungstest auch ein Test auf annähernde Übereinstimmung berechnet, der auf dem sog. “root mean square error of approximation” beruht. Faustregel: Der Wert des Koeffizienten RMSEA sollte kleiner als 0.05 sein, wenn die Modellanpassung noch als halbwegs befriedigend gelten soll.
CFI ist mit .976 fast sehr gut (.980 wäre sehr gut).
RMSEA mit 0.039 ebenfalls gut (der Wert sollte kleiner .05 sein, idealerweise auch der Wert des oberen Konfidenzintervalls, der hier 0.077 ist)
Für einen guten Modellfit sprechen:
CFI > .90 (Der CFI vergleicht das spezifizierte Modell mit dem theoretisch schlechtesten Modell) RMSEA < 0,05 (Der RMSEA zeigt an, wie gut sich die Daten mit Hilfe des spezifizierten Modell reproduzieren lassen. Je kleiner die Abweichungen, desto besser) RMSR < 0,10 (ähnlich wie der RMSEA, berücksichtigt jedoch nicht die Stichprobengröße für die Bewertung der Abweichung)
Obige Modellspezifikation fixiert automatisch die erste Ladung eines
Faktors auf 1.0 und lässt die Varianz des Faktors als zu schätzenden
Parameter frei. Der Nachteil ist, dass dadurch die Signifikanz der
ersten Ladung pro Faktor nicht getestet werden kann. Statt dessen kann
mit der Option “std.lv=T” festgelegt werden, dass die Varianzen der
Faktoren auf 1.0 fixiert werden und auch die ersten Ladungen der
Faktoren freie (d.h. zu schätzende) Parameter sind (geht nur, wenn die
latenten Variablen nicht abhängige Variablen sind):
m1_fit = sem(model1, data=ddf, std.lv=T)
summary(m1_fit, standardized=TRUE)
Vergleich eines ‘guten’ und eines ‘schlechten’ Modells mit
lavaan()
Idee: Die ‘richtige’ Zuordnung der Items zu den Faktoren wird verglichen mit einer ‘schlechten’, d. h. die ersten 6 Items und die zweiten 6 werden als je ein Faktor definiert.
Hierdurch können wir gut erkennen, wie sich die einzelnen Indices zueinander verhalten.
# get data
ddf <- read.delim("https://md.psych.bio.uni-goettingen.de/mv/data/virt/v_ei.txt")
head(ddf)## i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12
## 1 4 3 4 2 3 1 4 3 3 1 3 3
## 2 4 3 3 1 4 1 4 4 4 1 3 2
## 3 4 3 4 2 4 2 4 2 4 1 3 3
## 4 3 3 2 2 1 2 4 3 1 2 2 3
## 5 4 4 4 4 4 1 4 4 4 2 3 1
## 6 2 4 3 2 4 2 2 3 3 2 3 2require(lavaan)
# get cov matrix
#cov(ddf)
# and correlation matrix
#cor(ddf)
m.ei.good <- '
# EA Emotional Attention
EA =~ i1 + i3 + i5 + i7 + i9 + i11
# EC Emotional Clarity
EC =~ i2 + i4 + i6 + i8 + i10 + i12
'
# fit orthogonal good model
fit.g.o <- cfa(m.ei.good, data=ddf, orthogonal=T)
summary(fit.g.o, fit.measures=TRUE)## lavaan 0.6-11 ended normally after 27 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 24
##
## Number of observations 100
##
## Model Test User Model:
##
## Test statistic 69.883
## Degrees of freedom 54
## P-value (Chi-square) 0.072
##
## Model Test Baseline Model:
##
## Test statistic 397.852
## Degrees of freedom 66
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.952
## Tucker-Lewis Index (TLI) 0.942
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1440.660
## Loglikelihood unrestricted model (H1) -1405.718
##
## Akaike (AIC) 2929.320
## Bayesian (BIC) 2991.844
## Sample-size adjusted Bayesian (BIC) 2916.046
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.054
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.088
## P-value RMSEA <= 0.05 0.403
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.111
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## EA =~
## i1 1.000
## i3 0.918 0.234 3.932 0.000
## i5 0.773 0.230 3.358 0.001
## i7 0.465 0.209 2.230 0.026
## i9 0.976 0.246 3.961 0.000
## i11 0.705 0.192 3.675 0.000
## EC =~
## i2 1.000
## i4 -0.914 0.131 -6.981 0.000
## i6 -0.940 0.131 -7.154 0.000
## i8 0.987 0.142 6.970 0.000
## i10 -1.030 0.135 -7.649 0.000
## i12 -0.868 0.126 -6.904 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## EA ~~
## EC 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .i1 0.550 0.102 5.377 0.000
## .i3 0.481 0.088 5.438 0.000
## .i5 0.677 0.108 6.260 0.000
## .i7 0.776 0.114 6.817 0.000
## .i9 0.518 0.097 5.359 0.000
## .i11 0.398 0.067 5.917 0.000
## .i2 0.384 0.069 5.597 0.000
## .i4 0.454 0.075 6.034 0.000
## .i6 0.437 0.074 5.931 0.000
## .i8 0.532 0.088 6.040 0.000
## .i10 0.394 0.071 5.547 0.000
## .i12 0.426 0.070 6.076 0.000
## EA 0.306 0.114 2.684 0.007
## EC 0.550 0.127 4.313 0.000# fit good model with correlation between factors
fit.g.c <- cfa(m.ei.good, data=ddf, orthogonal=F)
summary(fit.g.c, fit.measures=TRUE)## lavaan 0.6-11 ended normally after 30 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 25
##
## Number of observations 100
##
## Model Test User Model:
##
## Test statistic 61.027
## Degrees of freedom 53
## P-value (Chi-square) 0.210
##
## Model Test Baseline Model:
##
## Test statistic 397.852
## Degrees of freedom 66
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.976
## Tucker-Lewis Index (TLI) 0.970
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1436.232
## Loglikelihood unrestricted model (H1) -1405.718
##
## Akaike (AIC) 2922.463
## Bayesian (BIC) 2987.593
## Sample-size adjusted Bayesian (BIC) 2908.636
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.039
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.077
## P-value RMSEA <= 0.05 0.642
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.055
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## EA =~
## i1 1.000
## i3 0.927 0.234 3.963 0.000
## i5 0.799 0.233 3.428 0.001
## i7 0.492 0.211 2.328 0.020
## i9 0.996 0.248 4.017 0.000
## i11 0.715 0.193 3.704 0.000
## EC =~
## i2 1.000
## i4 -0.904 0.130 -6.949 0.000
## i6 -0.932 0.130 -7.151 0.000
## i8 0.995 0.140 7.090 0.000
## i10 -1.029 0.133 -7.713 0.000
## i12 -0.861 0.125 -6.897 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## EA ~~
## EC 0.151 0.060 2.513 0.012
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .i1 0.560 0.101 5.549 0.000
## .i3 0.484 0.087 5.559 0.000
## .i5 0.670 0.107 6.268 0.000
## .i7 0.770 0.113 6.808 0.000
## .i9 0.516 0.095 5.431 0.000
## .i11 0.398 0.067 5.984 0.000
## .i2 0.381 0.068 5.602 0.000
## .i4 0.462 0.076 6.086 0.000
## .i6 0.441 0.074 5.973 0.000
## .i8 0.520 0.087 6.009 0.000
## .i10 0.391 0.070 5.555 0.000
## .i12 0.430 0.070 6.113 0.000
## EA 0.297 0.111 2.674 0.007
## EC 0.553 0.127 4.336 0.000# compare good model with bad model
m.ei.bad <- '
# item 1 to 6, diesregarding the content
f1 =~ i1 + i2 + i3 + i4 + i5 + i6
# item 7 to 12, diesregarding the content
f2 =~ i7 + i8 + i9 + i10 + i11 + i12
'
# fit orthogonal bad model
fit.b.o <- cfa(m.ei.bad, data=ddf, orthogonal=T)
summary(fit.b.o, fit.measures=TRUE)## lavaan 0.6-11 ended normally after 65 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 24
##
## Number of observations 100
##
## Model Test User Model:
##
## Test statistic 209.926
## Degrees of freedom 54
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 397.852
## Degrees of freedom 66
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.530
## Tucker-Lewis Index (TLI) 0.426
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1510.682
## Loglikelihood unrestricted model (H1) -1405.718
##
## Akaike (AIC) 3069.363
## Bayesian (BIC) 3131.887
## Sample-size adjusted Bayesian (BIC) 3056.089
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.170
## 90 Percent confidence interval - lower 0.146
## 90 Percent confidence interval - upper 0.194
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.230
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## f1 =~
## i1 1.000
## i2 2.799 1.329 2.105 0.035
## i3 0.634 0.516 1.229 0.219
## i4 -3.216 1.505 -2.138 0.033
## i5 0.605 0.538 1.125 0.261
## i6 -3.628 1.694 -2.142 0.032
## f2 =~
## i7 1.000
## i8 4.523 2.533 1.786 0.074
## i9 1.368 0.918 1.491 0.136
## i10 -3.841 2.158 -1.780 0.075
## i11 1.016 0.709 1.433 0.152
## i12 -3.221 1.823 -1.767 0.077
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## f1 ~~
## f2 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .i1 0.808 0.116 6.971 0.000
## .i2 0.553 0.095 5.797 0.000
## .i3 0.720 0.102 7.026 0.000
## .i4 0.410 0.091 4.514 0.000
## .i5 0.842 0.120 7.036 0.000
## .i6 0.282 0.097 2.918 0.004
## .i7 0.808 0.116 6.993 0.000
## .i8 0.372 0.112 3.320 0.001
## .i9 0.746 0.108 6.913 0.000
## .i10 0.475 0.099 4.787 0.000
## .i11 0.515 0.074 6.945 0.000
## .i12 0.487 0.087 5.601 0.000
## f1 0.049 0.045 1.088 0.277
## f2 0.034 0.038 0.904 0.366# fit bad model with correlation between factors
fit.b.c <- cfa(m.ei.bad, data=ddf, orthogonal=F)## Warning in lav_object_post_check(object): lavaan WARNING: covariance matrix of latent variables
## is not positive definite;
## use lavInspect(fit, "cov.lv") to investigate.summary(fit.b.c, fit.measures=TRUE)## lavaan 0.6-11 ended normally after 72 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 25
##
## Number of observations 100
##
## Model Test User Model:
##
## Test statistic 118.091
## Degrees of freedom 53
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 397.852
## Degrees of freedom 66
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.804
## Tucker-Lewis Index (TLI) 0.756
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1464.764
## Loglikelihood unrestricted model (H1) -1405.718
##
## Akaike (AIC) 2979.527
## Bayesian (BIC) 3044.657
## Sample-size adjusted Bayesian (BIC) 2965.700
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.111
## 90 Percent confidence interval - lower 0.084
## 90 Percent confidence interval - upper 0.138
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.109
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## f1 =~
## i1 1.000
## i2 3.523 1.631 2.159 0.031
## i3 1.045 0.633 1.651 0.099
## i4 -3.094 1.449 -2.135 0.033
## i5 1.099 0.674 1.631 0.103
## i6 -3.178 1.486 -2.140 0.032
## f2 =~
## i7 1.000
## i8 4.075 2.172 1.876 0.061
## i9 1.381 0.879 1.570 0.116
## i10 -4.151 2.202 -1.885 0.059
## i11 0.927 0.641 1.446 0.148
## i12 -3.444 1.842 -1.869 0.062
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## f1 ~~
## f2 0.039 0.028 1.416 0.157
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .i1 0.812 0.116 7.023 0.000
## .i2 0.383 0.069 5.555 0.000
## .i3 0.691 0.099 7.009 0.000
## .i4 0.489 0.079 6.222 0.000
## .i5 0.806 0.115 7.012 0.000
## .i6 0.474 0.077 6.137 0.000
## .i7 0.809 0.115 7.034 0.000
## .i8 0.514 0.086 5.999 0.000
## .i9 0.746 0.107 6.995 0.000
## .i10 0.402 0.072 5.587 0.000
## .i11 0.521 0.074 7.022 0.000
## .i12 0.445 0.072 6.206 0.000
## f1 0.044 0.041 1.088 0.277
## f2 0.033 0.035 0.948 0.343anova(fit.g.o, fit.g.c, test="Chisq")## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## fit.g.c 53 2922.5 2987.6 61.027
## fit.g.o 54 2929.3 2991.8 69.883 8.8564 1 0.002921 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(fit.b.o, fit.b.c, test="Chisq")## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## fit.b.c 53 2979.5 3044.7 118.09
## fit.b.o 54 3069.4 3131.9 209.93 91.836 1 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(fit.g.o, fit.b.o, test="Chisq")## Warning in lavTestLRT(object = object, ..., model.names = NAMES): lavaan
## WARNING: some models have the same degrees of freedom## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## fit.g.o 54 2929.3 2991.8 69.883
## fit.b.o 54 3069.4 3131.9 209.926 140.04 0anova(fit.g.c, fit.b.c, test="Chisq")## Warning in lavTestLRT(object = object, ..., model.names = NAMES): lavaan
## WARNING: some models have the same degrees of freedom## Chi-Squared Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## fit.g.c 53 2922.5 2987.6 61.027
## fit.b.c 53 2979.5 3044.7 118.091 57.064 0Die Indices im Vergleich (good - bad):
lavaan (0.5-18) converged normally after 30 iterations
Number of observations 100 100
Estimator ML
Minimum Function Test Statistic 61.027 118.091
Degrees of freedom 53 53
P-value (Chi-square) 0.210 0.000
Model test baseline model:
Minimum Function Test Statistic 397.852 397.852
Degrees of freedom 66 66
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.976 0.804
Tucker-Lewis Index (TLI) 0.970 0.756
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -1436.232 -1464.764
Loglikelihood unrestricted model (H1) -1405.718 -1405.718
Number of free parameters 25 25
Akaike (AIC) 2922.463 2979.527
Bayesian (BIC) 2987.593 3044.657
Sample-size adjusted Bayesian (BIC) 2908.636 2965.700
Root Mean Square Error of Approximation:
RMSEA 0.039 0.111
90 Percent Confidence Interval 0.000 0.077 0.084 0.138
P-value RMSEA <= 0.05 0.642 0.000
Standardized Root Mean Square Residual:
SRMR 0.055 0.109
Beispiel Emotionale Intelligenz mit library(sem)
Spezifikation des Modells Die Zeilen der Modellspezifikation müssen unmittelbar aufeinander folgen. Es darf keine Leerzeile dazwischen sein. Eine Leerzeile signalisiert das Ende der Spezifikation des Pfadmodells.
EA und EC sind latente Variablen EA -> i1, ea1 drückt
aus, dass die Ausprägung von i1 von der latenten Variable EA beeinflusst
wird. Dabei ist EA -> i1 der eigentliche path,
ea1 ist der Name des Pfades. EA <-> EA, NA, 1 EC
<-> EC, NA, 1 oder EA <-> EA, var_ea EC <-> EC, var_ec
definiert die Varianz der latenten Variablen
i1 <-> i1, error1 i2 <-> i2, error2 definiert bzw. deklariert die Varianz des Errors (disturbance parameter, error term)
EA <-> EC, cov1 deklariert einen Covarianz-Term. Notwendig, wenn eine oblique Struktur der Faktoren vermutet wird, also wenn Korrelationen zwischen den Faktoren zugelassen werden sollen.
Einschränkungen (constraints) definieren:
eine der beobachteten Variablen kann auf 1 gesetzt werden statt
EA -> NA, 1stattEA -> i1, ea1die Varianzen der latenten Variablen können auf 1 gesetzt werden
EA <-> EA, NA, 1stattEA <-> EA, var_ea
Die standardisierten Koeffizienten des Modells bekommt man über den
Befehl stdCoef(model) Die oben auf 1 gesetzten Varianzen
der latenten Variablen sind dann 1.
Sollen unterschiedliche Varianzen der latenten Variablen geschätzt
werden, kann man den Varianzen der latenten Variablen in der
Modellspezifikation einen Namen geben an Stelle von NA, 1
und dafür den jeweils ersten Pfad der beobachteten Variablen in ihre
latente Variable auf 1 setzen.
# get data
ddf <- read.delim("https://md.psych.bio.uni-goettingen.de/mv/data/virt/v_ei.txt")
head(ddf)## i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12
## 1 4 3 4 2 3 1 4 3 3 1 3 3
## 2 4 3 3 1 4 1 4 4 4 1 3 2
## 3 4 3 4 2 4 2 4 2 4 1 3 3
## 4 3 3 2 2 1 2 4 3 1 2 2 3
## 5 4 4 4 4 4 1 4 4 4 2 3 1
## 6 2 4 3 2 4 2 2 3 3 2 3 2cov.ei <- cov(ddf) ## covariance matrix to be used as the S argument in sem function
# we use library(sem)
require(sem)## Lade nötiges Paket: sem##
## Attache Paket: 'sem'## Die folgenden Objekte sind maskiert von 'package:lavaan':
##
## cfa, sem# EA and EC are names for latent variables in the path model
# EA -> i1, ea1 draw a path from latent variable EA to i1, which is influenced by EA
cfa.ei.good <- specifyModel(file="sem_ei_good.txt")## NOTE: it is generally simpler to use specifyEquations() or cfa()
## see ?specifyEquations# content of sem_ei_good.txt
# EA -> i1, ea1
# EA -> i3, ea2
# EA -> i5, ea3
# EA -> i7, ea4
# EA -> i9, ea5
# EA -> i11, ea6
# EC -> i2, ec1
# EC -> i4, ec2
# EC -> i6, ec3
# EC -> i8, ec4
# EC -> i10, ec5
# EC -> i12, ec6
# EA <-> EA, NA, 1
# EC <-> EC, NA, 1
# i1 <-> i1, error1
# i2 <-> i2, error2
# i3 <-> i3, error3
# i4 <-> i4, error4
# i5 <-> i5, error5
# i6 <-> i6, error6
# i7 <-> i7, error7
# i8 <-> i8, error8
# i9 <-> i9, error9
# i10 <-> i10, error10
# i11 <-> i11, error11
# i12 <-> i12, error12
# EA <-> EC, cov1
#
# compare it to a 'bad' model: first 6 items vs last 6 items
cfa.ei.bad <- specifyModel(file="sem_ei_bad.txt") ## NOTE: it is generally simpler to use specifyEquations() or cfa()
## see ?specifyEquations# content of sem_ei_bad.txt
# EA -> i1, ea1
# EA -> i2, ea2
# EA -> i3, ea3
# EA -> i4, ea4
# EA -> i5, ea5
# EA -> i6, ea6
# EC -> i7, ec1
# EC -> i8, ec2
# EC -> i9, ec3
# EC -> i10, ec4
# EC -> i11, ec5
# EC -> i12, ec6
# EA <-> EA, NA, 1
# EC <-> EC, NA, 1
# i1 <-> i1, error1
# i2 <-> i2, error2
# i3 <-> i3, error3
# i4 <-> i4, error4
# i5 <-> i5, error5
# i6 <-> i6, error6
# i7 <-> i7, error7
# i8 <-> i8, error8
# i9 <-> i9, error9
# i10 <-> i10, error10
# i11 <-> i11, error11
# i12 <-> i12, error12
# EA <-> EC, cov1
#
#
## nrow() function used to specify the number of observations.
cfa.ei.output.good <- sem(cfa.ei.good, cov.ei, nrow(ddf))
summary(cfa.ei.output.good)##
## Model Chisquare = 60.41636 Df = 53 Pr(>Chisq) = 0.2256253
## AIC = 110.4164
## BIC = -183.6577
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.97374 -0.17898 0.07269 0.16674 0.45324 1.85548
##
## R-square for Endogenous Variables
## i1 i3 i5 i7 i9 i11 i2 i4 i6 i8 i10
## 0.3464 0.3450 0.2204 0.0854 0.3634 0.2757 0.5921 0.4945 0.5212 0.5130 0.5996
## i12
## 0.4878
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## ea1 0.5474174 0.10288177 5.320840 1.032895e-07 i1 <--- EA
## ea2 0.5077084 0.09562707 5.309254 1.100749e-07 i3 <--- EA
## ea3 0.4375633 0.10539800 4.151533 3.302549e-05 i5 <--- EA
## ea4 0.2694231 0.10740247 2.508537 1.212322e-02 i7 <--- EA
## ea5 0.5452292 0.09982697 5.461742 4.714849e-08 i9 <--- EA
## ea6 0.3913831 0.08339001 4.693406 2.686940e-06 i11 <--- EA
## ec1 -0.7472475 0.08659951 -8.628772 6.201404e-18 i2 <--- EC
## ec2 0.6753677 0.08862688 7.620349 2.529910e-14 i4 <--- EC
## ec3 0.6966804 0.08823181 7.896023 2.879428e-15 i6 <--- EC
## ec4 -0.7437778 0.09521467 -7.811587 5.647213e-15 i8 <--- EC
## ec5 0.7690216 0.08832599 8.706628 3.130479e-18 i10 <--- EC
## ec6 0.6434796 0.08522406 7.550446 4.337709e-14 i12 <--- EC
## error1 0.5653847 0.10239907 5.521385 3.363383e-08 i1 <--> i1
## error2 0.3846516 0.06901060 5.573805 2.492348e-08 i2 <--> i2
## error3 0.4893029 0.08847064 5.530681 3.189897e-08 i3 <--> i3
## error4 0.4662017 0.07698670 6.055613 1.398836e-09 i4 <--> i4
## error5 0.6772253 0.10859775 6.236089 4.486453e-10 i5 <--> i5
## error6 0.4458486 0.07501455 5.943495 2.790090e-09 i6 <--> i6
## error7 0.7778152 0.11482072 6.774171 1.251213e-11 i7 <--> i7
## error8 0.5250775 0.08781776 5.979172 2.242741e-09 i8 <--> i8
## error9 0.5208059 0.09637789 5.403790 6.524720e-08 i9 <--> i9
## error10 0.3948685 0.07144268 5.527067 3.256293e-08 i10 <--> i10
## error11 0.4023748 0.06758147 5.953922 2.617916e-09 i11 <--> i11
## error12 0.4348229 0.07149183 6.082135 1.185929e-09 i12 <--> i12
## cov1 -0.3722188 0.11471502 -3.244726 1.175637e-03 EC <--> EA
##
## Iterations = 18summary(cfa.ei.output.good, fit.indices=c("RMSEA", "NNFI", "CFI"))##
## Model Chisquare = 60.41636 Df = 53 Pr(>Chisq) = 0.2256253
## RMSEA index = 0.03759586 90% CI: (NA, 0.07660595)
## Tucker-Lewis NNFI = 0.9718322
## Bentler CFI = 0.9773804
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.97374 -0.17898 0.07269 0.16674 0.45324 1.85548
##
## R-square for Endogenous Variables
## i1 i3 i5 i7 i9 i11 i2 i4 i6 i8 i10
## 0.3464 0.3450 0.2204 0.0854 0.3634 0.2757 0.5921 0.4945 0.5212 0.5130 0.5996
## i12
## 0.4878
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## ea1 0.5474174 0.10288177 5.320840 1.032895e-07 i1 <--- EA
## ea2 0.5077084 0.09562707 5.309254 1.100749e-07 i3 <--- EA
## ea3 0.4375633 0.10539800 4.151533 3.302549e-05 i5 <--- EA
## ea4 0.2694231 0.10740247 2.508537 1.212322e-02 i7 <--- EA
## ea5 0.5452292 0.09982697 5.461742 4.714849e-08 i9 <--- EA
## ea6 0.3913831 0.08339001 4.693406 2.686940e-06 i11 <--- EA
## ec1 -0.7472475 0.08659951 -8.628772 6.201404e-18 i2 <--- EC
## ec2 0.6753677 0.08862688 7.620349 2.529910e-14 i4 <--- EC
## ec3 0.6966804 0.08823181 7.896023 2.879428e-15 i6 <--- EC
## ec4 -0.7437778 0.09521467 -7.811587 5.647213e-15 i8 <--- EC
## ec5 0.7690216 0.08832599 8.706628 3.130479e-18 i10 <--- EC
## ec6 0.6434796 0.08522406 7.550446 4.337709e-14 i12 <--- EC
## error1 0.5653847 0.10239907 5.521385 3.363383e-08 i1 <--> i1
## error2 0.3846516 0.06901060 5.573805 2.492348e-08 i2 <--> i2
## error3 0.4893029 0.08847064 5.530681 3.189897e-08 i3 <--> i3
## error4 0.4662017 0.07698670 6.055613 1.398836e-09 i4 <--> i4
## error5 0.6772253 0.10859775 6.236089 4.486453e-10 i5 <--> i5
## error6 0.4458486 0.07501455 5.943495 2.790090e-09 i6 <--> i6
## error7 0.7778152 0.11482072 6.774171 1.251213e-11 i7 <--> i7
## error8 0.5250775 0.08781776 5.979172 2.242741e-09 i8 <--> i8
## error9 0.5208059 0.09637789 5.403790 6.524720e-08 i9 <--> i9
## error10 0.3948685 0.07144268 5.527067 3.256293e-08 i10 <--> i10
## error11 0.4023748 0.06758147 5.953922 2.617916e-09 i11 <--> i11
## error12 0.4348229 0.07149183 6.082135 1.185929e-09 i12 <--> i12
## cov1 -0.3722188 0.11471502 -3.244726 1.175637e-03 EC <--> EA
##
## Iterations = 18# summary output may be adapted using opt
opt <- options(fit.indices = c("GFI", "AGFI", "RMSEA", "NFI", "NNFI", "CFI", "RNI", "IFI", "SRMR", "AIC", "AICc", "BIC", "CAIC"))
summary(cfa.ei.output.good)##
## Model Chisquare = 60.41636 Df = 53 Pr(>Chisq) = 0.2256253
## Goodness-of-fit index = 0.9132819
## Adjusted goodness-of-fit index = 0.8723771
## RMSEA index = 0.03759586 90% CI: (NA, 0.07660595)
## Bentler-Bonett NFI = 0.8466097
## Tucker-Lewis NNFI = 0.9718322
## Bentler CFI = 0.9773804
## Bentler RNI = 0.9773804
## Bollen IFI = 0.9782431
## SRMR = 0.05505206
## AIC = 110.4164
## AICc = 77.98393
## BIC = -183.6577
## CAIC = -236.6577
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.97374 -0.17898 0.07269 0.16674 0.45324 1.85548
##
## R-square for Endogenous Variables
## i1 i3 i5 i7 i9 i11 i2 i4 i6 i8 i10
## 0.3464 0.3450 0.2204 0.0854 0.3634 0.2757 0.5921 0.4945 0.5212 0.5130 0.5996
## i12
## 0.4878
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## ea1 0.5474174 0.10288177 5.320840 1.032895e-07 i1 <--- EA
## ea2 0.5077084 0.09562707 5.309254 1.100749e-07 i3 <--- EA
## ea3 0.4375633 0.10539800 4.151533 3.302549e-05 i5 <--- EA
## ea4 0.2694231 0.10740247 2.508537 1.212322e-02 i7 <--- EA
## ea5 0.5452292 0.09982697 5.461742 4.714849e-08 i9 <--- EA
## ea6 0.3913831 0.08339001 4.693406 2.686940e-06 i11 <--- EA
## ec1 -0.7472475 0.08659951 -8.628772 6.201404e-18 i2 <--- EC
## ec2 0.6753677 0.08862688 7.620349 2.529910e-14 i4 <--- EC
## ec3 0.6966804 0.08823181 7.896023 2.879428e-15 i6 <--- EC
## ec4 -0.7437778 0.09521467 -7.811587 5.647213e-15 i8 <--- EC
## ec5 0.7690216 0.08832599 8.706628 3.130479e-18 i10 <--- EC
## ec6 0.6434796 0.08522406 7.550446 4.337709e-14 i12 <--- EC
## error1 0.5653847 0.10239907 5.521385 3.363383e-08 i1 <--> i1
## error2 0.3846516 0.06901060 5.573805 2.492348e-08 i2 <--> i2
## error3 0.4893029 0.08847064 5.530681 3.189897e-08 i3 <--> i3
## error4 0.4662017 0.07698670 6.055613 1.398836e-09 i4 <--> i4
## error5 0.6772253 0.10859775 6.236089 4.486453e-10 i5 <--> i5
## error6 0.4458486 0.07501455 5.943495 2.790090e-09 i6 <--> i6
## error7 0.7778152 0.11482072 6.774171 1.251213e-11 i7 <--> i7
## error8 0.5250775 0.08781776 5.979172 2.242741e-09 i8 <--> i8
## error9 0.5208059 0.09637789 5.403790 6.524720e-08 i9 <--> i9
## error10 0.3948685 0.07144268 5.527067 3.256293e-08 i10 <--> i10
## error11 0.4023748 0.06758147 5.953922 2.617916e-09 i11 <--> i11
## error12 0.4348229 0.07149183 6.082135 1.185929e-09 i12 <--> i12
## cov1 -0.3722188 0.11471502 -3.244726 1.175637e-03 EC <--> EA
##
## Iterations = 18## nrow() function used to specify the number of observations.
cfa.ei.output.bad <- sem(cfa.ei.bad, cov.ei, nrow(ddf))
summary(cfa.ei.output.bad)##
## Model Chisquare = 116.9098 Df = 53 Pr(>Chisq) = 1.025578e-06
## Goodness-of-fit index = 0.8160228
## Adjusted goodness-of-fit index = 0.7292411
## RMSEA index = 0.1103642 90% CI: (0.08332078, 0.1374512)
## Bentler-Bonett NFI = 0.7031793
## Tucker-Lewis NNFI = 0.7572667
## Bentler CFI = 0.8050778
## Bentler RNI = 0.8050778
## Bollen IFI = 0.8125116
## SRMR = 0.1087502
## AIC = 166.9098
## AICc = 134.4774
## BIC = -127.1642
## CAIC = -180.1642
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.7036 0.0000 0.4347 0.6330 1.0964 3.6355
##
## R-square for Endogenous Variables
## i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11
## 0.0518 0.5895 0.0655 0.4649 0.0623 0.4861 0.0396 0.5188 0.0785 0.5886 0.0521
## i12
## 0.4708
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## ea1 -0.2116710 0.09776987 -2.164992 3.038829e-02 i1 <--- EA
## ea2 -0.7456124 0.08696205 -8.573998 9.995324e-18 i2 <--- EA
## ea3 -0.2211681 0.09052516 -2.443167 1.455899e-02 i3 <--- EA
## ea4 0.6548425 0.08937066 7.327265 2.348965e-13 i4 <--- EA
## ea5 -0.2325893 0.09769995 -2.380649 1.728216e-02 i5 <--- EA
## ea6 0.6727726 0.08920159 7.542160 4.622503e-14 i6 <--- EA
## ec1 0.1835568 0.09730191 1.886467 5.923209e-02 i7 <--- EC
## ec2 0.7479376 0.09493449 7.878460 3.314410e-15 i8 <--- EC
## ec3 0.2534597 0.09443824 2.683867 7.277600e-03 i9 <--- EC
## ec4 -0.7619196 0.08883893 -8.576416 9.787543e-18 i10 <--- EC
## ec5 0.1700775 0.07838330 2.169818 3.002062e-02 i11 <--- EC
## ec6 -0.6321568 0.08554256 -7.389968 1.468640e-13 i12 <--- EC
## error1 0.8202460 0.11738694 6.987540 2.797470e-12 i1 <--> i1
## error2 0.3870925 0.07003767 5.526918 3.259055e-08 i2 <--> i2
## error3 0.6981555 0.10011064 6.973839 3.084075e-12 i3 <--> i3
## error4 0.4935045 0.07971769 6.190652 5.991586e-10 i4 <--> i4
## error5 0.8145891 0.11675208 6.977084 3.013686e-12 i5 <--> i5
## error6 0.4785889 0.07837148 6.106672 1.017300e-09 i6 <--> i6
## error7 0.8167110 0.11668896 6.999042 2.577180e-12 i7 <--> i7
## error8 0.5188724 0.08692543 5.969167 2.384683e-09 i8 <--> i8
## error9 0.7538390 0.10831432 6.959736 3.409114e-12 i9 <--> i9
## error10 0.4057410 0.07298967 5.558883 2.715068e-08 i10 <--> i10
## error11 0.5266294 0.07537432 6.986854 2.811182e-12 i11 <--> i11
## error12 0.4492666 0.07276221 6.174450 6.639439e-10 i12 <--> i12
## cov1 -1.0251487 0.04276360 -23.972464 5.389011e-127 EC <--> EA
##
## Iterations = 20summary(cfa.ei.output.bad, fit.indices=c("RMSEA"))##
## Model Chisquare = 116.9098 Df = 53 Pr(>Chisq) = 1.025578e-06
## RMSEA index = 0.1103642 90% CI: (0.08332078, 0.1374512)
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.7036 0.0000 0.4347 0.6330 1.0964 3.6355
##
## R-square for Endogenous Variables
## i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11
## 0.0518 0.5895 0.0655 0.4649 0.0623 0.4861 0.0396 0.5188 0.0785 0.5886 0.0521
## i12
## 0.4708
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## ea1 -0.2116710 0.09776987 -2.164992 3.038829e-02 i1 <--- EA
## ea2 -0.7456124 0.08696205 -8.573998 9.995324e-18 i2 <--- EA
## ea3 -0.2211681 0.09052516 -2.443167 1.455899e-02 i3 <--- EA
## ea4 0.6548425 0.08937066 7.327265 2.348965e-13 i4 <--- EA
## ea5 -0.2325893 0.09769995 -2.380649 1.728216e-02 i5 <--- EA
## ea6 0.6727726 0.08920159 7.542160 4.622503e-14 i6 <--- EA
## ec1 0.1835568 0.09730191 1.886467 5.923209e-02 i7 <--- EC
## ec2 0.7479376 0.09493449 7.878460 3.314410e-15 i8 <--- EC
## ec3 0.2534597 0.09443824 2.683867 7.277600e-03 i9 <--- EC
## ec4 -0.7619196 0.08883893 -8.576416 9.787543e-18 i10 <--- EC
## ec5 0.1700775 0.07838330 2.169818 3.002062e-02 i11 <--- EC
## ec6 -0.6321568 0.08554256 -7.389968 1.468640e-13 i12 <--- EC
## error1 0.8202460 0.11738694 6.987540 2.797470e-12 i1 <--> i1
## error2 0.3870925 0.07003767 5.526918 3.259055e-08 i2 <--> i2
## error3 0.6981555 0.10011064 6.973839 3.084075e-12 i3 <--> i3
## error4 0.4935045 0.07971769 6.190652 5.991586e-10 i4 <--> i4
## error5 0.8145891 0.11675208 6.977084 3.013686e-12 i5 <--> i5
## error6 0.4785889 0.07837148 6.106672 1.017300e-09 i6 <--> i6
## error7 0.8167110 0.11668896 6.999042 2.577180e-12 i7 <--> i7
## error8 0.5188724 0.08692543 5.969167 2.384683e-09 i8 <--> i8
## error9 0.7538390 0.10831432 6.959736 3.409114e-12 i9 <--> i9
## error10 0.4057410 0.07298967 5.558883 2.715068e-08 i10 <--> i10
## error11 0.5266294 0.07537432 6.986854 2.811182e-12 i11 <--> i11
## error12 0.4492666 0.07276221 6.174450 6.639439e-10 i12 <--> i12
## cov1 -1.0251487 0.04276360 -23.972464 5.389011e-127 EC <--> EA
##
## Iterations = 20# get standardized coefficients of good model
stdCoef(cfa.ei.output.good)## Std. Estimate
## 1 ea1 0.5885697 i1 <--- EA
## 2 ea2 0.5873994 i3 <--- EA
## 3 ea3 0.4694715 i5 <--- EA
## 4 ea4 0.2921610 i7 <--- EA
## 5 ea5 0.6028108 i9 <--- EA
## 6 ea6 0.5250956 i11 <--- EA
## 7 ec1 -0.7694876 i2 <--- EC
## 8 ec2 0.7032321 i4 <--- EC
## 9 ec3 0.7219536 i6 <--- EC
## 10 ec4 -0.7162701 i8 <--- EC
## 11 ec5 0.7743588 i10 <--- EC
## 12 ec6 0.6984082 i12 <--- EC
## 13 1.0000000 EA <--> EA
## 14 1.0000000 EC <--> EC
## 15 error1 0.6535857 i1 <--> i1
## 16 error2 0.4078889 i2 <--> i2
## 17 error3 0.6549619 i3 <--> i3
## 18 error4 0.5054646 i4 <--> i4
## 19 error5 0.7795965 i5 <--> i5
## 20 error6 0.4787830 i6 <--> i6
## 21 error7 0.9146420 i7 <--> i7
## 22 error8 0.4869571 i8 <--> i8
## 23 error9 0.6366192 i9 <--> i9
## 24 error10 0.4003685 i10 <--> i10
## 25 error11 0.7242746 i11 <--> i11
## 26 error12 0.5122259 i12 <--> i12
## 27 cov1 -0.3722188 EC <--> EA# compare the two models
cfa.ei.output.good##
## Model Chisquare = 60.41636 Df = 53
##
## ea1 ea2 ea3 ea4 ea5 ea6 ec1
## 0.5474174 0.5077084 0.4375633 0.2694231 0.5452292 0.3913831 -0.7472475
## ec2 ec3 ec4 ec5 ec6 error1 error2
## 0.6753677 0.6966804 -0.7437778 0.7690216 0.6434796 0.5653847 0.3846516
## error3 error4 error5 error6 error7 error8 error9
## 0.4893029 0.4662017 0.6772253 0.4458486 0.7778152 0.5250775 0.5208059
## error10 error11 error12 cov1
## 0.3948685 0.4023748 0.4348229 -0.3722188
##
## Iterations = 18cfa.ei.output.bad##
## Model Chisquare = 116.9098 Df = 53
##
## ea1 ea2 ea3 ea4 ea5 ea6 ec1
## -0.2116710 -0.7456124 -0.2211681 0.6548425 -0.2325893 0.6727726 0.1835568
## ec2 ec3 ec4 ec5 ec6 error1 error2
## 0.7479376 0.2534597 -0.7619196 0.1700775 -0.6321568 0.8202460 0.3870925
## error3 error4 error5 error6 error7 error8 error9
## 0.6981555 0.4935045 0.8145891 0.4785889 0.8167110 0.5188724 0.7538390
## error10 error11 error12 cov1
## 0.4057410 0.5266294 0.4492666 -1.0251487
##
## Iterations = 20# compare AIC and BIC
AIC(cfa.ei.output.good)## [1] 110.4164AIC(cfa.ei.output.bad)## [1] 166.9098BIC(cfa.ei.output.good)## [1] -183.6577BIC(cfa.ei.output.bad)## [1] -127.1642Vergleich der beiden Modelle nach Analyse mit SEM, good ist erster Wert, bad ist zweiter.
| Maß | good model | bad model |
|---|---|---|
| Model Chisquare | 60.41636 Df = 53 Pr(>Chisq) = 0.2256253 | 116.9098 Df = 53 Pr(>Chisq) = 1.025578e-06 |
| Goodness-of-fit index | 0.9132819 | 0.8160228 |
| Adjusted goodness-of-fit index | 0.8723771 | 0.7292411 |
| RMSEA index | 0.03759586 90% CI: (NA, 0.07660595) | 0.1103642 90% CI: (0.08332078, 0.1374512) |
| AIC | 110.4164 | 166.9098 |
| BIC | -183.6577 | -127.1642 |
- \(\chi^2\) Wert als Abweichungsmaß (reproduzierte Korrelationsmatrix von originaler) ist deutlich kleiner im good model und n. s.
- Goodness-of-fit ist höher im good model
- ebenso die adjusted goodness of fit
- RMSEA, als eine Art Residualanalyse ist deutlich kleiner im good model und liegt unter der Daumenregel-Grenze von 0.05
- AIC ist kleiner für good model. Je kleiner desto besser werden die Daten durch Modell erklärt
- BIC ist ebenfalls kleiner für good model. Kleiner ist hier “negativer”. Es zählt nicht der Absolutwert.
Weiteres Beispiel
Basis: Quelle
Inspired by the factor cookbook site
require(sem)
#write.table(validation.data, "cfa_validation.txt", sep="\t", quote=F, row.names=F)
# read ddf data
ddf <- read.delim("https://md.psych.bio.uni-goettingen.de/mv/data/div/cfa_validation.txt")
head(ddf)## V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 V17 V18
## 1 5 5 5 5 6 6 4 5 5 5 5 4 5 4 5 5 5 5
## 2 4 5 5 3 6 6 1 5 4 5 6 6 5 5 4 6 5 6
## 3 2 6 3 4 5 5 1 6 2 3 5 3 4 4 4 4 6 6
## 4 4 6 6 6 6 6 5 6 6 6 6 6 6 6 6 6 6 6
## 5 5 6 5 5 6 6 3 5 4 5 5 5 5 5 5 5 5 5
## 6 6 6 2 4 6 5 2 6 6 6 5 5 5 5 2 5 6 5cov.validation <- cov(ddf) ## covariance matrix to be used as the S argument in sem function
cov.validation## V1 V2 V3 V4 V5 V6
## V1 0.810010432 0.03811927 0.28889102 0.1670818 0.10086011 0.26370526
## V2 0.038119265 0.36573631 0.12949479 0.1371059 0.14269443 0.10476677
## V3 0.288891018 0.12949479 0.89089012 0.4654042 0.17451193 0.26856997
## V4 0.167081816 0.13710587 0.46540419 0.7766920 0.20302953 0.21428115
## V5 0.100860105 0.14269443 0.17451193 0.2030295 0.41785357 0.19834579
## V6 0.263705265 0.10476677 0.26856997 0.2142812 0.19834579 0.94204935
## V7 0.295123587 0.13890486 0.34264227 0.4296215 0.18237849 0.59048668
## V8 -0.007116093 0.19632859 0.07231057 0.1038300 0.14484469 0.05978157
## V9 0.534090183 0.05010538 0.23594878 0.1570703 0.11578953 0.25930361
## V10 0.202369547 0.08195482 0.23064231 0.1942049 0.03243491 0.19942092
## V11 0.168816930 0.10346810 0.24510336 0.1873124 0.15084307 0.48932852
## V12 0.234708650 0.03599029 0.22090226 0.1674437 0.06477401 0.25763769
## V13 0.249515659 0.07751591 0.36171255 0.3280641 0.17108429 0.39052820
## V14 0.269144791 0.13390177 0.32290136 0.1977923 0.10247812 0.26903834
## V15 0.224207490 0.08232739 0.37105342 0.3194684 0.12943625 0.21763961
## V16 0.256615784 0.11585872 0.23844500 0.2255328 0.15530859 0.69555683
## V17 -0.018277235 0.14339699 0.10044496 0.1537598 0.23825871 0.12747227
## V18 0.024355453 0.19496072 0.09674054 0.1029678 0.25311362 0.13978838
## V7 V8 V9 V10 V11 V12
## V1 0.2951236 -0.007116093 0.53409018 0.20236955 0.16881693 0.23470865
## V2 0.1389049 0.196328586 0.05010538 0.08195482 0.10346810 0.03599029
## V3 0.3426423 0.072310575 0.23594878 0.23064231 0.24510336 0.22090226
## V4 0.4296215 0.103830023 0.15707032 0.19420493 0.18731238 0.16744374
## V5 0.1823785 0.144844691 0.11578953 0.03243491 0.15084307 0.06477401
## V6 0.5904867 0.059781567 0.25930361 0.19942092 0.48932852 0.25763769
## V7 1.7996157 0.207596176 0.34935918 0.32489728 0.52293969 0.48742841
## V8 0.2075962 0.530662536 0.03075568 0.04049307 0.08351696 0.04106257
## V9 0.3493592 0.030755679 0.80054182 0.25314556 0.23391827 0.24480531
## V10 0.3248973 0.040493070 0.25314556 0.71469630 0.30167550 0.30865853
## V11 0.5229397 0.083516957 0.23391827 0.30167550 1.01681357 0.41356901
## V12 0.4874284 0.041062571 0.24480531 0.30865853 0.41356901 0.93161738
## V13 0.4859754 0.098363884 0.22933299 0.22331864 0.40483490 0.36238317
## V14 0.3650125 0.087809499 0.23261161 0.23562411 0.25619531 0.31977177
## V15 0.3186541 0.110166379 0.14560048 0.22918396 0.25496317 0.29037598
## V16 0.7338890 0.093999489 0.22307913 0.19320432 0.60068979 0.32631837
## V17 0.3090976 0.320743118 0.01292021 0.05667327 0.14630038 0.03940197
## V18 0.2540238 0.334280726 0.04649145 0.03496838 0.17363373 0.01073002
## V13 V14 V15 V16 V17 V18
## V1 0.24951566 0.26914479 0.22420749 0.25661578 -0.01827723 0.02435545
## V2 0.07751591 0.13390177 0.08232739 0.11585872 0.14339699 0.19496072
## V3 0.36171255 0.32290136 0.37105342 0.23844500 0.10044496 0.09674054
## V4 0.32806412 0.19779225 0.31946840 0.22553278 0.15375977 0.10296779
## V5 0.17108429 0.10247812 0.12943625 0.15530859 0.23825871 0.25311362
## V6 0.39052820 0.26903834 0.21763961 0.69555683 0.12747227 0.13978838
## V7 0.48597539 0.36501245 0.31865406 0.73388900 0.30909763 0.25402376
## V8 0.09836388 0.08780950 0.11016638 0.09399949 0.32074312 0.33428073
## V9 0.22933299 0.23261161 0.14560048 0.22307913 0.01292021 0.04649145
## V10 0.22331864 0.23562411 0.22918396 0.19320432 0.05667327 0.03496838
## V11 0.40483490 0.25619531 0.25496317 0.60068979 0.14630038 0.17363373
## V12 0.36238317 0.31977177 0.29037598 0.32631837 0.03940197 0.01073002
## V13 0.90647421 0.41902450 0.42440016 0.41716165 0.09321709 0.12878159
## V14 0.41902450 0.68725384 0.29506504 0.35716719 0.08622874 0.11906283
## V15 0.42440016 0.29506504 0.81204892 0.28296715 0.16900588 0.18931362
## V16 0.41716165 0.35716719 0.28296715 1.18351749 0.22747014 0.19639778
## V17 0.09321709 0.08622874 0.16900588 0.22747014 0.71437163 0.52819825
## V18 0.12878159 0.11906283 0.18931362 0.19639778 0.52819825 0.94709502attach(ddf)
## copy and paste this command separately into R before copying the model
cfa.validation <- sem::specifyModel(file="sem_model.txt")## NOTE: it is generally simpler to use specifyEquations() or cfa()
## see ?specifyEquations# just for documentation reasons. specifyModel read from stdin and this conflicts with KnitR
# content of sem_model.txt is (without comment #)
# ABILITY -> V12, ability0
# ABILITY -> V9, ability1
# ABILITY -> V14, ability2
# ABILITY -> V13, ability3
# ABILITY -> V3, ability4
# ABILITY -> V1, ability5
# ABILITY -> V15, ability6
# ABILITY -> V10, ability7
# VALUES -> V17, values0
# VALUES ->V18, values1
# VALUES -> V8, values2
# VALUES -> V2, values3
# VALUES -> V5, values4
# IDENTITY -> V16, identity0
# IDENTITY -> V6, identity1
# IDENTITY -> V11, identity2
# IDENTITY -> V7, identity3
# ABILITY <-> ABILITY, NA, 1
# VALUES <-> VALUES, NA, 1
# IDENTITY <-> IDENTITY, NA, 1
# V1 <-> V1, error1
# V2 <-> V2, error2
# V3 <-> V3, error3
# V4 <-> V4, error4
# V5 <-> V5, error5
# V6 <-> V6, error6
# V7 <-> V7, error7
# V8 <-> V8, error8
# V9 <-> V9, error9
# V10 <-> V10, error10
# V11 <-> V11, error11
# V12 <-> V12, error12
# V13 <-> V13, error13
# V14 <-> V14, error14
# V15 <-> V15, error15
# V16 <-> V16, error16
# V17 <-> V17, error17
# V18 <-> V18, error18
# ABILITY <-> VALUES, cov1
# ABILITY <-> IDENTITY, cov2
# VALUES <-> IDENTITY, cov3
#
## model specified using standardised factor variances. Analysis has also been run after setting the first item score for each factor to 1, with no difference
## line numbers for the model have been omitted for ease of copying and pasting into R
## nrow() function used to specify the number of observations.
cfa.validation.output <- sem::sem(cfa.validation, cov.validation, nrow(ddf))
# more informations are to be obtained
# See the section "ML.methods" in sem.pdf
?summary.objectiveML
#summary(X,fit.indices=c("RMSEA",...))
summary(cfa.validation.output, fit.indices=c("RMSEA"))##
## Model Chisquare = 561.2528 Df = 133 Pr(>Chisq) = 5.854301e-54
## RMSEA index = 0.1025802 90% CI: (NA, NA)
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -2.51237 -0.43183 0.02767 0.66295 1.47155 9.78711
##
## R-square for Endogenous Variables
## V12 V9 V14 V13 V3 V1 V15 V10 V17 V18 V8
## 0.3193 0.2699 0.4813 0.4904 0.3310 0.3021 0.3544 0.2525 0.6333 0.5825 0.4169
## V2 V5 V16 V6 V11 V7
## 0.2248 0.3106 0.6653 0.5932 0.4485 0.3899
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## ability0 0.5454256 0.05495730 9.924534 3.256189e-23 V12 <--- ABILITY
## ability1 0.4648402 0.05171841 8.987906 2.519863e-19 V9 <--- ABILITY
## ability2 0.5751229 0.04485033 12.823158 1.216427e-37 V14 <--- ABILITY
## ability3 0.6667419 0.05135888 12.982018 1.547491e-38 V13 <--- ABILITY
## ability4 0.5430359 0.05354916 10.140887 3.637813e-24 V3 <--- ABILITY
## ability5 0.4946864 0.05151662 9.602464 7.805609e-22 V1 <--- ABILITY
## ability6 0.5364778 0.05075407 10.570143 4.098707e-26 V15 <--- ABILITY
## ability7 0.4247777 0.04912394 8.647061 5.284253e-18 V10 <--- ABILITY
## values0 0.6726096 0.04487096 14.989865 8.552626e-51 V17 <--- VALUES
## values1 0.7427623 0.05225037 14.215445 7.348274e-46 V18 <--- VALUES
## values2 0.4703353 0.04077475 11.534966 8.792193e-31 V8 <--- VALUES
## values3 0.2867428 0.03579227 8.011306 1.134969e-15 V2 <--- VALUES
## values4 0.3602499 0.03731974 9.653065 4.770800e-22 V5 <--- VALUES
## identity0 0.8873503 0.05543298 16.007622 1.130485e-57 V16 <--- IDENTITY
## identity1 0.7475428 0.05048877 14.806122 1.337368e-49 V6 <--- IDENTITY
## identity2 0.6753142 0.05482191 12.318327 7.217620e-35 V11 <--- IDENTITY
## identity3 0.8376139 0.07429317 11.274439 1.754934e-29 V7 <--- IDENTITY
## error1 0.5652955 0.04986735 11.335985 8.704746e-30 V1 <--> V1
## error2 0.2835150 0.02444977 11.595816 4.327216e-31 V2 <--> V2
## error3 0.5960018 0.05327544 11.187177 4.711963e-29 V3 <--> V3
## error4 0.7766920 0.06279183 12.369317 3.830654e-35 V4 <--> V4
## error5 0.2880738 0.02581887 11.157491 6.582297e-29 V5 <--> V5
## error6 0.3832292 0.04263115 8.989418 2.485441e-19 V6 <--> V6
## error7 1.0980209 0.10041134 10.935227 7.820970e-28 V7 <--> V7
## error8 0.3094475 0.02970430 10.417601 2.060859e-25 V8 <--> V8
## error9 0.5844651 0.05087751 11.487691 1.521236e-30 V9 <--> V9
## error10 0.5342599 0.04619898 11.564324 6.248167e-31 V10 <--> V10
## error11 0.5607651 0.05324925 10.530948 6.220486e-26 V11 <--> V11
## error12 0.6341278 0.05637253 11.248880 2.345511e-29 V12 <--> V12
## error13 0.4619288 0.04592463 10.058410 8.434950e-24 V13 <--> V13
## error14 0.3564872 0.03515096 10.141605 3.611160e-24 V14 <--> V14
## error15 0.5242402 0.04741430 11.056583 2.037115e-28 V15 <--> V15
## error16 0.3961271 0.05073686 7.807481 5.834244e-15 V16 <--> V16
## error17 0.2619686 0.03471455 7.546364 4.475775e-14 V17 <--> V17
## error18 0.3954005 0.04696524 8.419004 3.796997e-17 V18 <--> V18
## cov1 0.2758005 0.06547343 4.212403 2.526678e-05 VALUES <--> ABILITY
## cov2 0.6920402 0.04301632 16.087854 3.104127e-58 IDENTITY <--> ABILITY
## cov3 0.3573852 0.06216556 5.748926 8.981225e-09 IDENTITY <--> VALUES
##
## Iterations = 30detach(ddf)CFA mit library(lavaan)
# this is included in the above mentioned documentation
# load the lavaan package (only needed once per session)
require(lavaan)
# specify the model
HS.model <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9 '
# fit the model
fit <- lavaan::cfa(HS.model, data=HolzingerSwineford1939)
# display summary output
summary(fit, fit.measures=TRUE)## lavaan 0.6-11 ended normally after 35 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 21
##
## Number of observations 301
##
## Model Test User Model:
##
## Test statistic 85.306
## Degrees of freedom 24
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 918.852
## Degrees of freedom 36
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.931
## Tucker-Lewis Index (TLI) 0.896
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -3737.745
## Loglikelihood unrestricted model (H1) -3695.092
##
## Akaike (AIC) 7517.490
## Bayesian (BIC) 7595.339
## Sample-size adjusted Bayesian (BIC) 7528.739
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.092
## 90 Percent confidence interval - lower 0.071
## 90 Percent confidence interval - upper 0.114
## P-value RMSEA <= 0.05 0.001
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.065
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## visual =~
## x1 1.000
## x2 0.554 0.100 5.554 0.000
## x3 0.729 0.109 6.685 0.000
## textual =~
## x4 1.000
## x5 1.113 0.065 17.014 0.000
## x6 0.926 0.055 16.703 0.000
## speed =~
## x7 1.000
## x8 1.180 0.165 7.152 0.000
## x9 1.082 0.151 7.155 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## visual ~~
## textual 0.408 0.074 5.552 0.000
## speed 0.262 0.056 4.660 0.000
## textual ~~
## speed 0.173 0.049 3.518 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .x1 0.549 0.114 4.833 0.000
## .x2 1.134 0.102 11.146 0.000
## .x3 0.844 0.091 9.317 0.000
## .x4 0.371 0.048 7.779 0.000
## .x5 0.446 0.058 7.642 0.000
## .x6 0.356 0.043 8.277 0.000
## .x7 0.799 0.081 9.823 0.000
## .x8 0.488 0.074 6.573 0.000
## .x9 0.566 0.071 8.003 0.000
## visual 0.809 0.145 5.564 0.000
## textual 0.979 0.112 8.737 0.000
## speed 0.384 0.086 4.451 0.000todo: adapt
require(lavaan)
model <- '
# latent variable definitions
factor_1 =~ y1 + y2 + y3 + y4
factor_2 =~ y5 + y6 + y7 + y8
# covariance between factor_1 and factor_2
factor_1 ~~ factor_2
# residual covariances
y1 ~~ y5
'
#fit <- cfa(model, data=ex_data)
#summary(fit)
# if we have a group variable and want a multiple-group confirmatory factor analysis to test for measurement invariance.
#measurement.invariance(model, data=ex_data, group ="school" )Weitere Beispiele, Übungen / Exercises
Beispiel BFI (Big Five Inventory)
Beispiel BFI
Links und Referenzen
lavaan cheat-sheet
Bemerkungen zu Fit-Indices mit ihrer Definition.
UCLA Video Tutorial
Timothy A. Brown:
Confirmatory Factor Analysis for Applied Research datasets, syntax files
at the companion
website
Field (2012, p. 112) Kapitel 3: R Environment